Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 22-23, 2016 Contest Prep - Geometry Today we will be focusing on how to solve geometry-based questions which come up on the math contests. Before we jump right in, let s review some basic angles/shape properties: 1 Angles Some Patterns Straight line a + b = 180 X pattern a = d C pattern c + e = 180 Z pattern d = e F pattern d = h Full circle a + b + c + d = 360 1
Angles within a polygon 1. What is the sum of the interior angles in a square? 360 2. What is the sum of the interior angles in a triangle? 180 3. What is the sum of the interior angles in a rectangle? 360 In general, do you see a pattern? Turns out there is! The sum of the interior angles within an n-sided polygon is: 180 (n 2) If you were to plug in n = 3 (for a triangle) or n = 4 (for a square or rectangle), what should the sum of the interior angles be? Does this line up with what we got before? How can knowing this help us when solving geometric problems? Example: Find the missing angles. (a) a = 180 50 2 = 130 2 = 65 (b) x = 360 70 80 110 = 100 2
(c) 180 (6 2) = 180 4 = 720 y = 720 270 50 130 110 110 = 50 (d) 180 (7 2) = 180 5 = 900 z = 900 130 110 130 130 130 110 = 160 (e) (This is a regular star-polygon. Solve for the rest of the angles.) 180 (10 2) = 180 8 = 1440 = 1440 (36 5) = 1260 = 1260 5 = 252 3
2 Area and Perimeter Polygon Area Perimeter s A = s 2 P = 4s s w A = l w P = 2l + 2w l a c h A = 1 2 bh P = a + b + c b b a h c A = h b + d 2 P = a + b + c + d d a h A = h b P = 2a + 2b b r A = πr 2 P = 2πr 4
3 Word Problems 1. What is the perimeter and area of the figure shown? (Gauss 7, 2006, Question 4) Note that the height of each triangle is not 2. Rather by Pythagorean Theorem, it is 22 1 2 = 4 1 = 3. P = 2 8 = 16 units A = (2 A square ) + (2 A triangle ) = 2 (2 2) + 2 ( 1 2 2 3) = 8 + 2 3 11.46 2. What is the area of the figure, in square units? (Gauss 7, 2010, Question 11) Solution 1: A = (2 3) + [5 (6 + 2)] = 6 + (5 8) = 6 + 40 = 46 Solution 2: A = [2 (3 + 5)] + (5 6) = (2 8) + 30 = 6 + 40 = 46 5
3. P QR has an area of 27cm 2 and a base measuring 6 cm. What is the height, h, of P QR. (Gauss 7, 2009, Question 12) 1 2 6 h = 27 h = 27 2 6 h = 9 cm 4. A square has an area of 25. A rectangle has the same width as the square. The length of the rectangle is double its width. What is the area of the rectangle? (Gauss 7, 2009, Question 16) Since the area of the square is 25 cm 2, we know that the width of the square is 5 cm, and thus the dimensions of the rectangle is 5 cm 10 cm. So the area of the rectangle is 5 10 cm 2 = 10 cm 2. 5. Each of P QR and ST U has an area of 1. in P QR, U, W and V are the midpoints of the sides, as shown. In ST U, R, V and W are the midpoints of the sides. What is the area of parallelogram UV RW? (Gauss 7, 2009, Question 20) Since V is the midpoint of P R, then P V = V R. Since UVRW is a parallellogram, then V R = UW. Since W is the midpoint of US, then UW = W S. Thus P V = V R = UW = W S. Similarily, QW = W R = UV = V T. 6
Also, r is the midpoint of T S and therefore, T R = RS. Thus, V T R is congruent to W RS, and so the two triangles have equal area. Diagonal V W in parallelogram UV RW divides the area of the parallelogram in half. Therefore, UV W and RW V have equal areas. In quadrilateral V RSW, V R = W S and V R is parallel to W S. Thus, V RSW is a parallelogram and the area of RW V is equal to the area of W RS. Therefore, V T R, W RS, RW V, and UV W have equal areas, and so these four triangles divide ST U into quarters. Parallelogram UV RW is made from two of these four quarters of ST U, or one half of ST U. The area of parallelogram UV RW is thus 1 of 1, or 1. 2 2 6. In the diagram shown, ST UV is a square, Q and P are the midpoints of ST and UV, P R = QR, and V Q is parallel to P R. What is the ratio of the shaded area to the unshaded area? (Gauss 7, 2014, Question 24) We begin by joining Q to P. Since Q and P are the midpoints of ST and UV, then QP is parallel to both SV and T U and rectangles SQP V and QT UP are identical. In rectangle SQP V, V Q is a diagonal. Similarly, since P R is parallel to V Q then P R extended to T is a diagonal of rectangle QT UP, as shown in Figure 1. In Figure 2, we label points A, B, C, D, E, and F, the midpoints of SQ, QT, T U, UP, P V, and V S, respectively. 7
We join A to E, B to D and F to C, with F C intersecting QP at the centre of the square O, as shown. Since P R = QR and R lies on diagonal P T, then both F C and BD pass through R. (That is, R is the centre of QT UP.) The line segments AE, QP, BD, and F C divide square ST U V into 8 identical rectangles. In one of these rectangles, QBRO, diagonal QR divides the rectangle into 2 equal areas. That is, the area of QOR is half of the area of rectangle QBRO. Similarly, the area of P OR is half of the area of rectangle P ORD. We begin by joining Q to P. Since Q and P are the midpoints of ST and UV, then QP is parallel to both SV and T U and rectangles SQP V and QT UP are identical. In rectangle SQP V, V Q is a diagonal. Similarly, since P R is parallel to V Q then P R extended to T is a diagonal of rectangle QT UP, as shown in Figure 1. In Figure 2, we label points A, B, C, D, E, and F, the midpoints of SQ, QT, T U, UP, P V, and V S, respectively. We join A to E, B to D and F to C, with F C intersecting QP at the centre of the square O, as shown. Since P R = QR and R lies on diagonal P T, then both F C and BD pass through R. (That is, R is the centre of QT UP.) The line segments AE, QP, BD, and F C divide square ST UV into 8 identical rectangles. In one of these rectangles, QBRO, diagonal QR divides the rectangle into 2 equal areas. That is, the area of QOR is half of the area of rectangle QBRO. Similarly, the area of P OR is half of the area of rectangle P ORD. We join A to E, B to D and F to C, with F C intersecting QP at the centre of the square O, as shown. Since P R = QR and R lies on diagonal P T, then both F C and BD pass through R. (That is, R is the centre of QT UP.) The line segments AE, QP, BD, and F C divide square ST UV into 8 identical rectangles. In one of these rectangles, QBRO, diagonal QR divides the rectangle into 2 equal areas. That is, the area of QOR is half of the area of rectangle QBRO. Similarly, the area 8
of P OR is half of the area of rectangle P ORD. Rectangle SQP V has area equal to 4 of the 8 identical rectangles. Therefore, QP V has area equal to 2 of the 8 identical rectangles (since diagonal V Q divides the area of SQP V in half). Thus the total shaded are, which is QOR + P OR + QP V, is equivalent to the area of 1 + 1 + 2 = 3 of the the identical rectangles. 2 2 Since square ST UV is divided into 8 of these identical rectangles, and the shaded area is equivalent to the area of 3 of these 8 rectangles, then the unshaded area aoccupies an area equal to that of the remaining 8 3 = 5 rectangles. Therefore, the ratio of the shaded area to the unshaded area is 3 : 5. 7. The area of square ABCD is 64 and AX = BW = CZ = DY = 2. What is the area of square W XY Z? (Gauss 7, 2004, Question 20) Solution 1: Since the area of square ABCD is 64, then the side length of square ABCD is 8. Since AX = BW = CZ = DY = 2, then AW = BZ = CY = DX = 6. Thus, each of triangles XAW, W BZ, ZCY and Y DX is right-angled with one leg of length 2 and the other of length 6. Therefore, each of these four triangles has area 1 2 6 = 6. Therefore, the area of 2 square W XY Z is equal to the area of square ABCD minus the sum of the areas of the four triangles, or 64 (4 6) = 40. Solution 2: Since the area of square ABCD is 64, then the side length of square ABCD is 8. Since AX = BW = CZ = DY = 2, then AW = BZ = CY = DX = 6. By Pythagorean Theorem, XW = W Z = ZY = Y X = 2 2 + 6 2 = 4 + 36 = 40. Therefore, the area of the square W XY Z is ( 40) 2 = 40. 9
4 Problem Set 1. In the diagram, the rectangle has length 11 and width 7. What is the area of the shaded part? (Problems, Problems, Problems, Volume 6: page 13, question 6) The unshaded triangle and the rectangle have the same base and the same height. Hence, the area of that traingle is 1 the area of the rectangle. 2 Notice that the shaded area is the other half of the rectangle. Therefore, the shaded area is 1(11)(7) = 77 = 38.5 2 2 units2. 2. The rectangle in the diagram has length 10 cm and height 8 cm. What is the area of the shaded part? (Problems, Problems, Problems, Volume 6: page 13, question 8) The shaded area is the area of the rectangle minus the area of the unshaded triangle. The area of the rectangle is 8 10 = 80 cm 2, The area of the triangle is 1 10 4 = 20 2 cm2. Therefore, the shaded area is 80 20 = 60 cm 2. 3. A garden, 10 m 20 m, is enclosed by a sidewalk of width 1 m. What is the area of the sidewalk? (Problems, Problems, Problems, Volume 6: page 13, question 10) Solutions 1: The area formed by the sidewalk and the garden is a rectangle with dimensions 12 m by 22 m. The area of the sidewalk is the area of the large rectangle minus the area of the garden. Therefore the area is (12 22) (10 20) = 264 200 = 64 cm 2. Solution 2: The area of the sidewalk may be considered as four rectangles, one along each edge of the garden, plus the four squares at the corners. The total area is (10 1) + (20 1) + (10 1) + (20 1) + (4 1) = 10 + 20 + 10 + 20 + 4 = 64 cm 2. 10
4. Of the five figures shown, which one has the greatest perimeter? (Problems, Problems, Problems, Volume 6: page 12, question 4) The perimeter of each of the firgures A, C, D and E is 10 units. The perimeter of figure B is 12 units. Hence, figure B has the greatest perimeter. 5. The areas of the two squares inside ABCD are 4 cm 2 and 9 cm 2. What is the area of the shaded area? (Problems, Problems, Problems, Volume 6: page 14, question 11) Solution 1: Since the 2 squares have areas of 4 cm 2 and 9 cm 2, the sides of the 2 squares are 2 cm and 3 cm, respectively. Therefore, the dimensions of each of the rectangles is 2 cm by 3 cm. Therefore, the total shaded area is (3 2) + (3 2) = 12 cm 2. Solution 2: Since the 2 squares have areas of 4 cm 2 and 9 cm 2, the sides of the 2 smalled squares are 2 cm and 3 cm, respectively. Therefore, the side of the square ABCD is 5 cm. The side of the square ABCD is 5 cm. The shaded area is the area of square ABCD minus the sum of the areas of the 2 smaller squares. Thus, the shaded area is 5 2 (4 + 9) = 25 13 = 12 cm 2. 11
6. What is ABD? (Problems, Problems, Problems, Volume 6: page 40, question 3) Solution 1: Since ADC is a straight angle, ADB + BDC = 180 ADB = 180 119 = 61 Therefore, ABD = 180 62 61 = 57 Solution 2: In BDC, C = 180 119 18 = 43. Therefore, ABC = 180 43 62 = 75. Hence, ABD = 75 18 = 57. 7. In the diagram, what is the value of x? (Problems, Problems, Problems, Volume 6: page 40, question 4) Since BOC and F OE are oposite angles, BOC = 30. AOB + BOC + COD = 180 110 + 30 + y = 180 = y = 40. 12
8. In the diagram, what is CAD? (Problems, Problems, Problems, Volume 6: page 43, question 3) Since BCD is a straight angle, ACD = 180 120 = 60 Similirily, ADC = 180 110 = 70. In ACD, the sum of the 3 interior angles is 180. Thus, CAD + 60 + 70 = 180. Hence, CAD = 180 130 = 50. 9. In the diagram, A = 40. Find the size of DBC.(Problems, Problems, Problems, Volume 6: page 44, question 4) Since AD = DB, ADB is isosceles and so DAB = DBA = 40. Therefore, ADB = 180 40 40 = 100. since ADC is a straight angle, BDC = 180 100 = 80. Since BDC is isosceles, BDC = BCD = 80. Therefore, DBC = 180 80 80 = 20. 13
10. One angle in a triangle is twice the size of the second angle and the third is 66. What is the smallest angle? (Problems, Problems, Problems, Volume 6: page 39, question 2) Then, 2x + x + 66 = 180 3x = 114 x = 38 The smallest angle is 38. 11. One angle in a triangle is 120, and the second is five times the third. What is the third angle? (Problems, Problems, Problems, Volume 6: page 43, question 13) 120 + y + 5y = 180 6y = 60 y = 10. Therefore, the third angle is 10. 12. In parallelogram ACEF, what is the value of CBD? (Problems, Problems, Problems, Volume 7: page 34, question 9) Since opposite angles in a parallelogram are equal, ACE = 80. The angle supplementary to 150 is 30, so BDC = 30. In BCD : x + 80 + 30 = 180. Therefore, x = 70. 14
13. In right triangle ABC, AX = AD and CY = CD, as shown. What is the measure of XDY? (Problems, Problems, Problems, Volume 7: page 37, question 12) Let DY C be x. Then Y DC = x because Y DC is isosceles. Since the sum of the interior angles in a triangle is 180, Y CD = (180 2x). In ABC, B = 90 and ACB = (180 2x). Therefore BAC = 180 90 (180 2x) = (2x 90). In ADX, AXD = ADX. Thereofre, ADX = 2[ 1 180 (2x 90) ] = (135 x), and XDY = 180 ADX CDY = 180 (135 x) x = 45. 14. A beam of light shines from point S, reflects off a reflector MN at point P, and reaches point T so that P T is perpendicular to RS. Given that MSP = 26, what is the measure of SP M. (Problems, Problems, Problems, Volume 7: page 37, question 10) In the diagram extend T P to meet RS at A. Since P AM = 90, then: SP A = 180 90 26 = 64. The X-pattern tells us that T P N = MP A = x. So, SP A = 2x = 64. Therefore SP M = x = 32. 15